﻿956 
  Prof. 
  A. 
  H. 
  Gibson 
  on 
  the 
  Motion 
  of 
  

  

  cylindrical 
  air 
  column 
  of 
  radius 
  r, 
  surrounded 
  by 
  an 
  inviscid 
  

   liquid 
  whose 
  boundaries 
  are 
  remote, 
  the 
  relationship 
  between 
  

   the 
  wave-length 
  I, 
  and 
  the 
  velocity 
  v 
  of 
  flow 
  past 
  the 
  surface, 
  

   is 
  given 
  by 
  

  

  where 
  &= 
  — 
  . 
  

  

  A 
  few 
  preliminary 
  measurements 
  of 
  wave-length 
  showed 
  

   that 
  for 
  all 
  the 
  tubes 
  in 
  which 
  the 
  series 
  of 
  ripples 
  was 
  well 
  

   defined 
  the 
  value 
  of 
  kr 
  was 
  large, 
  ranging 
  from 
  a 
  minimum 
  

   of 
  about 
  40 
  in 
  the 
  shortest 
  bubbles 
  in 
  the 
  tube 
  *788 
  cm. 
  

   diameter, 
  to 
  over 
  600 
  in 
  the 
  tube 
  5'09 
  cm. 
  diameter. 
  For 
  

   large 
  values 
  of 
  kr, 
  the 
  function 
  <j)(kr) 
  in 
  formula 
  (2) 
  is 
  

   defined 
  by 
  

  

  Wr)=-{2kr) 
  e 
  I 
  1 
  " 
  1^07) 
  + 
  1.2. 
  (Mr)' 
  

  

  l 
  2 
  . 
  3 
  2 
  . 
  o 
  2 
  

  

  1.2.3. 
  (Skr)' 
  

  

  ...}, 
  

  

  while, 
  writing 
  $(Jcr)-7-<f>{kr) 
  as 
  F(kr) 
  } 
  calculations 
  show 
  

   that 
  ¥(kr) 
  has 
  values 
  ranging 
  from 
  approximately 
  —1*01 
  

   when 
  £r 
  = 
  40, 
  to 
  — 
  1*001 
  when 
  At 
  =600. 
  

   "Writing 
  (2) 
  as 
  

  

  e 
  .= 
  _F(ir){i-^}^ 
  > 
  . 
  ... 
  (3) 
  

  

  it 
  appears 
  that 
  the 
  error 
  involved 
  in 
  taking 
  ¥(kr) 
  = 
  — 
  1 
  and 
  

  

  in 
  neglecting 
  the 
  additive 
  term 
  -=-^ 
  is, 
  for 
  these 
  experiments, 
  

  

  in 
  no 
  case 
  greater 
  than 
  1 
  per 
  cent., 
  and 
  is, 
  in 
  the 
  great 
  

   majority 
  of 
  cases, 
  much 
  less 
  than 
  this. 
  Since 
  this 
  is 
  in 
  

   general 
  not 
  greater 
  than 
  the 
  errors 
  of 
  measurement, 
  these 
  

   simplifications 
  have 
  been 
  adopted 
  in 
  the 
  calculations. 
  

   Equation 
  (3) 
  then 
  becomes 
  

  

  *-**-*£,■ 
  (i) 
  

  

  -37 
  

  

  the 
  formula 
  for 
  the 
  velocity 
  of 
  ripples 
  on 
  a 
  flat 
  sheet 
  of 
  

   deep 
  water, 
  with 
  only 
  surface-tension 
  operative. 
  Where 
  the 
  

   depth 
  of 
  water 
  " 
  h 
  " 
  is 
  not 
  great 
  comparable 
  with 
  the 
  wave- 
  

   length, 
  it 
  may 
  readily 
  be 
  shown 
  that 
  (4) 
  becomes 
  

  

  *- 
  toh 
  H' 
  f££\ 
  

  

  m 
  « 
  

  

  * 
  Sci. 
  Papers, 
  Kayleigli, 
  vol. 
  iii. 
  p. 
  595. 
  

  

  